This paper constructs an exactly solvable holographic model of de Sitter quantum gravity from the {4,3,3} tessellation of S³. Eight cubes tile the three-sphere; placing a perfect tensor at each cell, contracting across shared faces, and designating one cell as the observer produces a holographic model in 3+1 dimensions. Every result is exact: the Ryu-Takayanagi entropy formula holds with zero bulk entanglement, the de Sitter entropy is S = 6 log χ, the Hilbert space dimension is χ⁶ (confirming Banks's finite-dimensionality conjecture as a theorem), the reduced state is maximally mixed (reproducing the Gibbons-Hawking temperature), the Page curve is reproduced exactly, and the observer algebra is the full matrix algebra Mat(χ⁶) with a unique trace state. The proofs use only the perfect tensor property and the graph structure of the tessellation. No approximation, no large-N limit, and no holographic dictionary beyond the tensor network itself is required.
This paper constructs an exactly solvable holographic model of de Sitter quantum gravity from the {4,3,3} tessellation of S³. Eight cubes tile the three-sphere; placing a perfect tensor at each cell, contracting across shared faces, and designating one cell as the observer produces a holographic model in 3+1 dimensions. Every result is exact: the Ryu-Takayanagi entropy formula holds with zero bulk entanglement, the de Sitter entropy is S = 6 log χ, the Hilbert space dimension is χ⁶ (confirming Banks's finite-dimensionality conjecture as a theorem), the reduced state is maximally mixed (reproducing the Gibbons-Hawking temperature), the Page curve is reproduced exactly, and the observer algebra is the full matrix algebra Mat(χ⁶) with a unique trace state. The proofs use only the perfect tensor property and the graph structure of the tessellation. No approximation, no large-N limit, and no holographic dictionary beyond the tensor network itself is required.
Holographic tensor networks serve as toy models for the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, capturing many of its essential features in a concrete manner. However, existing holographic tensor network models remain far from a complete theory of quantum gravity. A key obstacle is their discrete structure, which only approximates the semi-classical geometry of gravity in a qualitative sense. In \cite{Lin:2024dho}, it was shown that a network of partial-entanglement-entropy (PEE) threads, which are bulk geodesics with a specific density distribution, generates a perfect tessellation of AdS space. Moreover, such PEE-network tessellations can be constructed for more highly symmetric geometries using the Crofton formula. In this paper, we assign a quantum state to each vertex in the PEE network and develop several holographic tensor network models: (1) the factorized PEE tensor network, which takes the form of a tensor product of EPR pairs; (2) the HaPPY-like PEE tensor network constructed from perfect tensors; and (3) the random PEE tensor network. In all these models, we reproduce the exact Ryu-Takayanagi formula by showing that the minimal number of cuts along a homologous surface in the network exactly equals the area of that surface.
This paper establishes the geometric and entropic foundations of an exactly solvable holographic model for two-dimensional quantum gravity. Taking the binary tree perfect tensor network — a hierarchical quantum system with N=2LN = 2^L N=2L boundary sites and bond dimension χ\chi χ — as its setting, it proves three main results: First, an exact closed-form formula S(Am)=f∗(m)logχS(A_m) = f^*(m)\log\chi S(Am)=f∗(m)logχ is derived for the entanglement entropy of any boundary interval, where f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m))f^*(m) = \min(\mathrm{popcount}_2(m),\ 1 + \mathrm{popcount}_2(N_{\min}(m)-m)) f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m)) is a combinatorial function of the interval size, establishing the Ryu–Takayanagi formula as a theorem of the network rather than a conjecture. Second, the tree is shown to embed isometrically into the hyperbolic plane H2\mathbb{H}^2 H2 with every edge of equal hyperbolic length η=arccosh(21/16)\eta = \mathrm{arccosh}(21/16) η=arccosh(21/16), and the rescaled boundary metric is proved quasi-isometric to AdS2\mathrm{AdS}_2 AdS2 geodesics with explicit tight constants. Third, Newton's constant G=1/(4logχ)G = 1/(4\log\chi) G=1/(4logχ) and the AdS2\mathrm{AdS}_2 AdS2 radius R=δ/ln2R = \delta/\ln 2 R=δ/ln2 are derived independently from two routes — the RT formula and the embedding geometry respectively — yielding R/G=4log2χR/G = 4\log_2\chi R/G=4log2χ as a proved theorem, with no free parameters once χ\chi χ is specified. The paper additionally derives the exact modular Hamiltonian of the vacuum state from the perfect tensor isometry, and recovers the bulk stress-energy tensor from boundary entropy perturbations via an explicit inversion formula, completing the chain from a discrete quantum network to a bulk metric with gravitational coupling and matter content.
We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with nonintersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.
The relation between loop quantum gravity (LQG) and tensor networks is explored from the perspectives of the bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin-network states in a space $\mathrm{\ensuremath{\Sigma}}$ with boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$ is an exact holographic mapping similar to the proposal in [X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282]. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse-graining procedure of spin networks. Furthermore, when a region A and its complement A are specified on the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$, we show that the boundary entanglement entropy S(A) of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime, i.e., S(A) is proportional to the minimal area of the bulk surface attached to the boundary of A in $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$.
Extending the holographic principle to de Sitter (dS) spacetimes remains one of the most vital open frontiers in quantum gravity, where a microscopic, bottom-up tensor-network framework that relates boundary quantum data to emergent de Sitter spacetime is still lacking. In this work, we first show the emergence of de Sitter spacetime from boundary entanglement by formulating a non-unitary continuous multi-scale entanglement renormalization ansatz (cMERA) for a concrete non-Hermitian critical fermion chain. Within this emergent spacetime, we analyze the associated geodesics and show that they act as extremal Ryu-Takayanagi (RT) surfaces undergoing a smooth timelike-to-null transition. Remarkably, we demonstrate that this continuum trajectory suggests a distinct tensor-network architecture in which the bond-counting contribution naturally truncates at the discrete timelike-to-null transition toward the deep infrared. In the resulting architecture, the null ray along the horizon is represented by zero-cost links, since the associated cut severs no tensor legs. This network structure successfully reproduces the logarithmic scaling of non-unitary critical entanglement entropy, offering a bond-counting picture for the de Sitter RT formula. Our results provide the long-sought dS/(c)MERA correspondence at the level of both emergent spacetime and discrete holographic entanglement.
A bstract We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large- N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with non intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.